How and why to think about scattering in terms of wave packets instead of plane waves

We discuss “the plane wave approximation” to quantum mechanical scattering using simple onedimensional examples. Our central claims are that (a) the plane waves of standard calculations can and should be thought of as very wide wave packets, and (b) the calculations of reflection and transmission probabilities R and T in standard textbook presentations involve an approximation which is almost never discussed. We present a simple and intuitively revealing alternative way to derive and understand the connection between asymptotic wave function amplitudes and scattering probabilities, which also has the benefit of bringing the approximate character of standard plane wave calculations out into the light. We then develop an under-appreciated exact expression for scattering probabilities, using it to calculate, for two standard examples, expressions for R and T for an incident wave packet. Comparing these results to the corresponding probabilities calculated using the plane wave approximation helps illuminate the domain of applicability of that approximation and thus underscores the importance of thinking about scattering in terms of wave packets instead of plane waves.